Disjoint p$p$‐convergent operators and their adjoints

First, we give conditions on a Banach lattice E$E$ so that an operator T$T$ from E$E$ to any Banach space is disjoint p$p$‐convergent if and only if T$T$ is almost Dunford–Pettis. Then, we study when adjoints of positive operators between Banach lattices are disjoint p$p$‐convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices E$E$ and F$F$: (i) a positive operator T:E→F$T: E \rightarrow F$ is almost weak p$p$‐convergent if and only if T∗$T^*$ is disjoint p$p$‐convergent; (ii) E∗$E^*$ has order continuous norm or F∗$F^*$ has the positive Schur property of order p$p$. Very recent results are improved, examples are given and applications of the main results are provided.